Simplify and expand the following expression: $ \dfrac{3}{5y - 10}- \dfrac{2}{5y - 30}- \dfrac{1}{y^2 - 8y + 12} $
Answer: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $5$ out of denominator in the first term: $ \dfrac{3}{5y - 10} = \dfrac{3}{5(y - 2)}$ We can factor a $5$ out of denominator in the second term: $ \dfrac{2}{5y - 30} = \dfrac{2}{5(y - 6)}$ We can factor the quadratic in the third term: $ \dfrac{1}{y^2 - 8y + 12} = \dfrac{1}{(y - 2)(y - 6)}$ Now we have: $ \dfrac{3}{5(y - 2)}- \dfrac{2}{5(y - 6)}- \dfrac{1}{(y - 2)(y - 6)} $ The least common multiple of the denominators is: $ 25(y - 2)(y - 6)$ In order to get the first term over $25(y - 2)(y - 6)$ , multiply by $\dfrac{5(y - 6)}{5(y - 6)}$ $ \dfrac{3}{5(y - 2)} \times \dfrac{5(y - 6)}{5(y - 6)} = \dfrac{15(y - 6)}{25(y - 2)(y - 6)} $ In order to get the second term over $25(y - 2)(y - 6)$ , multiply by $\dfrac{5(y - 2)}{5(y - 2)}$ $ \dfrac{2}{5(y - 6)} \times \dfrac{5(y - 2)}{5(y - 2)} = \dfrac{10(y - 2)}{25(y - 2)(y - 6)} $ In order to get the third term over $25(y - 2)(y - 6)$ , multiply by $\dfrac{25}{25}$ $ \dfrac{1}{(y - 2)(y - 6)} \times \dfrac{25}{25} = \dfrac{25}{25(y - 2)(y - 6)} $ Now we have: $ \dfrac{15(y - 6)}{25(y - 2)(y - 6)} - \dfrac{10(y - 2)}{25(y - 2)(y - 6)} - \dfrac{25}{25(y - 2)(y - 6)} $ $ = \dfrac{ 15(y - 6) - 10(y - 2) - 25} {25(y - 2)(y - 6)} $ Expand: $ = \dfrac{15y - 90 - 10y + 20 - 25}{25y^2 - 200y + 300} $ $ = \dfrac{5y - 95}{25y^2 - 200y + 300}$ Simplify: $ = \dfrac{y - 19}{5y^2 - 40y + 60}$